On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs

Abstract

It is shown that whenever the edges of a connected simple graph on n vertices are colored with $n-1$ colors appearing so that no cycle in G is rainbow, there must be a monochromatic edge cut in G. From this it follows that such colorings of G can be represented, or ‘encoded,’ by full binary trees with n leaves, with vertices labeled by subsets of V(G), such that the leaf labels are singletons, the label of each non-leaf is the union of the labels of its children, and each label set induces a connected subgraph of G. It is also shown that $n-1$ is the largest integer for which the main theorem holds, for each n, although for some graphs a certain strengthening of the hypothesis makes the theorem conclusion true with $n-1$ replaced by $n-2$.